This seems really neat, but it seems quite sensitive to how one defines the worlds under consideration, and whether one counts slightly different worlds as actually distinct. Let me try to illustrate this with an example.

Suppose we have a consisting of 7 worlds, , with preferences

and no other non-trivial preferences. Then (from the `sensible case’), I think we get the following utilities:

.

Suppose now that I create two new copies , of the world which each differ by the position of a single atom, so as to give me (extremely weak!) preferences , so all the non-trivial preferences in the new are now summarised as

Then the resulting utilities are (I think):

.

In particular, before adding in these ‘trivial copies’ we had , and now we get . Is this a problem? It depends on the situation, but to me it suggests that, if using this approach, one needs to be careful in how the worlds are specified, and the ‘fine-grainedness’ needs to be roughly the same everywhere.

Are you sure? Suppose we take W=W1⊔W2 with W1={A,B,C,D}, W2={X,Y,Z}, then n1=3, so the values for W1 should be −3,−1,1,3 as I gave them. And similarly for W2, giving values −2,0,2. Or else I have mis-understood your definition?

Just to be clear, by “separate partial preference” you mean a separate preorder, on a set of objects which may or may not have some overlap with the objects we considered so far? Then somehow the work is just postponed to the point where we try to combine partial preferences?

EDIT (in reply to your edit): I guess e.g. keeping conditions 1,2,3 the same and instead minimising

g(G)=∑w←w′λw←w′(U(w′)−U(w))2,

where λw←w′∈R>0 is proportion to the reciprocal of the strength of the preference? Of course there are lots of variants on this!